2. WHAT IS IT?

As discussed here by
Kolb (2003),
particle candidates for dark matter range from the axion with a mass
10-15
GeV
(van Bibber 2003)
to cryptons with masses
10+15
GeV
(Ellis et al. 1990;
Benakli et al.
1999),
via neutrinos with masses
10-10
GeV, the gravitino and the
lightest supersymmetric particle with a mass
102 GeV
(Ellis et al. 1984;
Goldberg 1983).
In recent years, there has been considerable
experimental progress in understanding neutrino masses, so I start with
them, even though cosmology now disfavours the hot dark matter they would
provide
(Spergel et al.
2003).
All the others are candidates for cold dark matter,
except for the gravitino, which might constitute warm dark matter, another
possibility now disfavoured by the WMAP evidence for reionization when
z ~ 20
(Kogut et al. 2003).

Particle theorists expect particles to have masses that vanish exactly
only if they are protected by some unbroken gauge symmetry, much as the
photon is massless because of the U(1) gauge symmetry of electromagnetism,
that is associated with the conservation of electric charge. There is no
corresponding exact gauge symmetry to protect lepton number, so we expect
it to be violated and neutrinos to acquire masses. This is indeed the
accepted interpretation of the observed oscillations between different
types of neutrinos, which are made possible by mixing into non-degenerate
mass eigenstates
(Wark 2003);
Pakvasa & Valle 2003).

Neutrino masses could arise even within the Standard Model of particle
physics, without adding any new particles, at the expense of introducing a
interaction between two neutrino fields and two Higgs
fields
(Barbieri et al.
1980):

(2.1)

However, such an interaction would be non-renormalizable, and therefore
is not thought
to be fundamental. The (presumably large) mass scale M appearing the
denominator of (2.1) is generally thought to originate from the
exchange of some massive fermionic particle that mixes with the light
neutrino
(Gell-Mann et al.
1979;
Yanagida 1979;
Mohapatra & Senjanovic
1980):

(2.2)

Diagonalization of this matrix naturally yields small neutrino masses,
since we expect that the Dirac mass term mD is of the
same order as quark and lepton masses, and M >>
mW.

and there are prospects to improve this limit by a factor ~ 20.
From measurements of n decay, we know that
(Hagiwara et al.
2002):

(2.5)

and there are prospects to improve this limit to ~ 5 MeV.

However, the most stringent laboratory limit on neutrino masses may come
from searches for neutrinoless
double-
decay, which constrain the sum of the neutrinos' Majorana masses
weighted by their couplings to electrons
(Klapdor-Kleingrothaus et
al. 2001):

(2.6)

and there are prospects to improve this limit to ~ 0.01 eV in a
future round of experiments. The impact of the limit (2.6) in
relation to the cosmological upper limit (1.1) is
discussed below, after we have gathered further experimental input from
neutrino-oscillation experiments.

The neutrino mass matrix (2.2) should be regarded also as a
matrix in flavour space. When it is diagonalized, the neutrino mass
eigenstates will not, in general, coincide with the flavour eigenstates
that partner the mass eigenstates of the charged leptons. The mixing
matrix between them
(Maki et al. 1962)
may be written in the form

(2.7)

where the symbols c, sij denote the standard
trigonometric functions of the three real `Euler' mixing angles
12,23,31, and
is a
CP-violating phase that can in principle also be observed in
neutrino-oscillation experiments
(De Rújula et
al. 1999).
Additionally, there are two CP-violating phases
1,2
that appear in the
double-
observable (2.6), but do not affect neutrino oscillations.

The pioneering Super-Kamiokande and other experiments have shown that
atmospheric neutrinos oscillate, with the following difference in squared
masses and mixing angle
(Fukuda et al.
1998):

(2.8)

which is very consistent with the K2K reactor neutrino experiment
(Ahn et al. 2002),
as seen in the left panel of Fig. 6.
A flurry of recent solar neutrino experiments, most notably
SNO (Ahmad et al.
2002a,
b),
have established beyond any doubt that they also oscillate, with

(2.9)

Most recently, the KamLAND experiment has reported a deficit of electron
antineutrinos from nuclear power reactors, leading to a very similar set
of preferred parameters, as seen in the
right panel of Fig. 6
(Eguchi et al.
2003).

Using the range of
12 allowed by
the solar and KamLAND data,
one can establish a correlation between the relic neutrino density
h2 and
the neutrinoless
double- decay
observable <m>e, as seen in Fig. 7
(Minakata & Sugiyama
2002).
Pre-WMAP, the experimental limit on
<m>e could be used to set the bound
(Minakata & Sugiyama
2002)

(2.10)

Alternatively, now that WMAP has set a tighter upper bound
h2 <
0.0076 (1.1), one can use this correlation to set an upper bound:

Figure 7. Correlation between
h2 and <m>e. The
different diagonal lines correspond to uncertainties in the measurements
by KamLAND and other experiments
(Minakata & Sugiyama
2002).

The `Holy Grail' of neutrino physics is CP violation in neutrino
oscillations, which would manifest itself as a difference between the
oscillation probabilities for neutrinos and antineutrinos
(De Rújula et
al. 1999):

(2.12)

For this to be observable,
m12
and 12 have
to be large, as SNO and KamLAND have shown to be the case, and also
13
has to be large enough - which remains to be seen.

In fact, even the minimal seesaw model contains many additional parameters
that are not observable in neutrino oscillations
(Casas & Ibarra 2001).
In addition to the three masses of the charged leptons, there are three
light-neutrino masses, three light-neutrino mixing angles and three
CP-violating phases in the light-neutrino sector: the oscillation phase
and the two Majorana
phases that are relevant to neutrinoless
double- decay
experiments. As well, there are three heavy
singlet-neutrino masses, three more mixing angles and three more
CP-violating phases that become observable in the heavy-neutrino sector,
making a total of 18 parameters. Out of all these parameters, so far just
four light-neutrino parameters are known, two differences in masses
squared and two real mixing angles.

As discussed later, it is often thought that there may be a connection
between CP violation in the neutrino sector and the baryon density in the
Universe, via leptogenesis. Unfortunately, this connection is somewhat
indirect, since the three CP-violating phases measurable in the
light-neutrino sector to not contribute to leptogenesis, which is
controlled by the other phases that are not observable directly at low
energies
(Ellis & Raidal 2002).
However, if the seesaw model is combined
with supersymmetry, these extra phases contribute to the renormalization
of soft superymmetry-breaking parameters at low energies, and hence may
have indirect observable effects
(Ellis et al. 2002a,
b).